An hour was a measurement of time, usually equal to 60 minutes. (DTI novel: Watching the Clock)
Originating as an Earth measure of time by the Human civilization, hours were used throughout the Federation. In the year 2285, Captain Spock used a simple code of converting hours to days in order to mislead Khan Noonien Singh, who he suspected was eavesdropping communications. Rear Admiral James T. Kirk correctly ascertained Spock's meaning based on a reference to Saavik's "by the book" behavior of following Starfleet regulations. (TOS movie & novelization: The Wrath of Khan)